Cross-over trials

Steve Simon

Topics to be covered

  • What you will learn
    • The paired t-test
    • The AB/BA cross-over
    • Period effects and carry-over
    • When should you use or not use a cross-over
    • Cross-over trials with a binary outcome
    • Cross-over trials with more than two treatments

Quotations from two great philosophers

  • “A man cannot step into the same river twice, because it is not the same river, and he is not same man.” - Heraclitus, 544 B.C.
  • “You must unlearn what you have learned.” Yoda, 1980 A.D.

Paired t-test data

Descriptive statistics on wife’s age

variable mean sd n
wAge 40.7 11.4 170

Descriptive statistics on husband’s age

variable mean sd n
hAge 42.6 11.6 199

Descriptive statistics on age gap

variable mean sd n
age_gap 2.2 4.1 170

Paired t-test on couples ages

  • Difference mean (\(\bar{D}\)) = 2.2
  • Difference standard deviation (\(S_D\)) = 4.1
  • Difference standard error (\(S_D / \sqrt{n}\)) = 0.3
  • Test statistic \(\Big(\frac{\bar{D}}{S_D/\sqrt{n}}\Big)\) = 7.15
  • p<0.001

Analyze as completely randomized block design

Analysis of Variance Table

Response: age
           Df Sum Sq Mean Sq F value    Pr(>F)    
spouse      1    425  424.71  51.148 2.474e-11 ***
pair      169  43974  260.20  31.337 < 2.2e-16 ***
Residuals 169   1403    8.30                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Restructured data for a completely randomized block design

Mixed model/random coefficient analysis

  • Uses long format
  • Dependent variable: age
  • Fixed effect: spouse
  • Random intercept: pair

Break #1

  • What you have learned
    • The paired t-test
  • What’s coming next
    • The AB/BA cross-over

Parallel groups versus the AB/BA cross-over

https://statistically-funny.blogspot.com/2021/03/in-clinical-trials-you-can-have-it-both.html

Data layout

Alternate data layout

Descriptive statistics, period 1

variable mean1 sd n
period1 312.3 82.8 13
1 Larger values imply better outcomes

Descriptive statistics, period 2

variable mean1 sd n
period2 324.6 67.9 13
1 Larger values imply better outcomes

Descriptive statistics, period gap

variable mean1 sd n
period_gap 12.3 61.0 13
1 Defined as period2 - period1

Descriptive statistics, formoterol

variable mean1 sd n
formoterol 341.2 59.7 13
1 Larger values imply better outcomes

Descriptive statistics, salbutamol

variable mean1 sd n
salbutamol 295.8 82.9 13
1 Larger values imply better outcomes

Descriptive statistics, drug gap

variable mean1 sd n
drug_gap 45.4 40.6 13
1 Defined as formoterol - salbutamol

Paired t-test for the AB/BA cross-over data

  • Difference mean (\(\bar{D}\)) = 45.4
  • Difference standard deviation (\(S_D\)) = 40.6
  • Difference standard error (\(S_D/\sqrt{n}\)) = 11.3
  • Test statistic \(\Big(\frac{\bar{D}}{S_D/\sqrt{n}}\Big)\) = 4.03
  • p<0.001

Analysis using a completely randomized block design

Analysis of Variance Table

Response: pef
          Df Sum Sq Mean Sq F value    Pr(>F)    
treatment  1  13388 13388.5  16.250  0.001666 ** 
id        12 115213  9601.1  11.654 7.935e-05 ***
Residuals 12   9887   823.9                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Break #2

  • What you have learned
    • The AB/BA cross-over
  • What’s coming next
    • Period effects and carry-over

Accounting for period effects

  • Improves precision
  • Adjusts for imbalances
  • Requires layout used for completely randomized block design

Why might you see period effects

  • Temporal trends
  • Seasonal effects
  • Carry-over: Persistence of a treatment applied in one period in a subsequent period of treatment
    • Physical persistence
    • Curative effect

Layout for asthma data

Counts for layout data

treatment order period n
formoterol for/sal 1 7
formoterol sal/for 2 6
salbutamol for/sal 2 7
salbutamol sal/for 1 6

Means, 1

# A tibble: 2 × 3
  order   period1_mean period2_mean
  <chr>          <dbl>        <dbl>
1 for/sal         337.         306.
2 sal/for         283.         346.

Interaction, 1

What this graph would look like without a period effect

Means, 2

# A tibble: 2 × 3
  order   formoterol_mean salbutamol_mean
  <chr>             <dbl>           <dbl>
1 for/sal            337.            306.
2 sal/for            346.            283.

Interaction, 2

Accounting for period effects

Analysis of Variance Table

Response: pef
          Df Sum Sq Mean Sq F value    Pr(>F)    
order      1    335   335.2  0.4467  0.517697    
id        11 114878 10443.5 13.9171 6.495e-05 ***
period     1    985   984.6  1.3121  0.276323    
treatment  1  14036 14035.9 18.7044  0.001205 ** 
Residuals 11   8254   750.4                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Break #3

  • What you have learned
    • Period effects and carry-over
  • What’s coming next
    • When should you use or not use a cross-over

Systematic overview of parachute use

When you cannot use a cross-over trial

  • Acute conditions
  • Mortality/morbidity
  • Training/education
  • Irreversible treatments
  • Persistent effects

When you can use a cross-over trial

  • Chronic conditions
  • Short-term benefits
  • Adequate wash-out period
    • Drugs with a short half-life

Ethical concerns

  • Can you ask a patient to endure two treatments?
  • Can you leave the patient untreated during the wash-out period

What is you have a carry-over effect?

  • No easy answer
    • Adjustments have poor power
    • Between subject comparison
  • Unclear where to make adjustments
    • Two active treatments:
      • Split the difference
    • Active treatment versus placebo:
      • Assign the carry-over to placebo
    • Carry-over could an interaction instead

Change test if carry-over is detected

  • Test hypothesis of no carry-over
    • If accept null, run the cross-over
    • If reject null, look at first period only
  • Not recommended
    • Limited power to test carry-over
    • First period analysis loses to much power/precision

Break #4

  • What you have learned
    • When should you use or not use a cross-over
  • What’s coming next
    • Cross-over trials with a binary outcome

Example of a binary outcome

Descriptive statistics

F- F+
S- 2 15
S+ 1 6

Analysis of binary cross-over data

  • Ignoring period effects
    • McNemar’s test
  • Incorporating period effects
    • Mainland-Gart test (hard to find)

Example of McNemar’s test

Break #5

  • What you have learned
    • Cross-over trials with a binary outcome
  • What’s coming next
    • Cross-over trials with more than two treatments

Three treatment cross-over trial

  • Show example data from page 124 of Senn

Possible sequences

  • Three treatments
    • ABC, ACB, BAC, BCA, CAB, CBA
  • Four treatments
    • ABCD, ACBD, BACD, BCAD, CABD, CBAD,
    • ABDC, ACDB, BADC, BCDA, CADB, CBAD,
    • ADBC, ADCB, BDAC, BDCA, CDAB, CDBA,
    • DABC, DACB, DBAC, DBCA, DCAB, DCBA

Latin square design to reduce the number of sequences

  • Three treatments
    • Sequence 1: A B C
    • Sequence 2: B C A
    • Sequence 3: C A B
  • Four treatments
    • Sequence 1: A B C D
    • Sequence 2: B C D A
    • Sequence 3: C D A B
    • Sequence 4: D A B C

The definitive guide

Nice reference

Summary

  • What you have learned
    • The paired t-test
    • The AB/BA cross-over
    • Period effects and carry-over
    • When should you use or not use a cross-over
    • Cross-over trials with a binary outcome
    • Cross-over trials with more than two treatments